10.4.1. Comparing very large and very small numbers

The diameter of the Sun is 1.4 × 10^9 m and the diameter of the Earth is 1.2756 × 10^7 m. Suppose you want to compare the diameter of the Earth, with the diameter of the Sun. Diameter of the Sun = 1.4 × 10^9 m Diameter of the earth = 1.2756 × 10^7 m 1.4 × 10^9 / 1.2756 × 10^7 = (1.4 / 1.2756) × 10^(9 - 7) Therefore, approximately = 1.1 × 10^2 = 100. So, the diameter of the Sun is about 100 times the diameter of the Earth.

To compare the diameters of the Sun and the Earth, we express both diameters in scientific notation, which makes it easier to do calculations. The diameter of the Sun is given as 1.4 × 10^9 meters and the diameter of the Earth is 1.2756 × 10^7 meters. We divide the two values, ensuring to handle the powers of ten correctly by subtracting the exponents. This results in approximately 100, indicating that the Sun's diameter is about 100 times larger than the Earth's.

Think of it like comparing two swimming pools. If one pool has a diameter of 1.4 kilometers and another pool has a diameter of 0.12756 kilometers, it’s clear that the first pool is much larger. Just like how we would compare the sizes of these pools, we can compare the sizes of celestial bodies using powers of ten.

Let us compare the size of a Red Blood cell which is 0.000007 m to that of a plant cell which is 0.00001275 m. Size of Red Blood cell = 0.000007 m = 7 × 10^−6 m Size of plant cell = 0.00001275 = 1.275 × 10^−5 m. 7 × 10^−6 / 1.275 × 10^−5 = (0.7 / 1.275)
Therefore, a red blood cell is half of a plant cell in size.

Comparing the sizes of a Red Blood cell and a plant cell involves converting both measurements into scientific notation. The size of the Red Blood cell is 0.000007 m (7 × 10^−6 m) and the plant cell is 0.00001275 m (1.275 × 10^−5 m). By dividing these two values, we understand that the size of a Red Blood cell is roughly half the size of a plant cell, providing a clearer perspective on their comparative sizes.

Imagine trying to compare the sizes of two very tiny objects, like two different grains of sand. If one grain is 7 micrometers wide and the other is about 12.75 micrometers, it's easier to understand that one is smaller than the other if we use the idea of dividing their sizes, just like we did with the calculations for the cells.

Mass of Earth is 5.97 × 10^24 kg and mass of Moon is 7.35 × 10^22 kg. What is the total mass? Total mass = 5.97 × 10^24 kg + 7.35 × 10^22 kg. When we have to add numbers in standard form, we convert them into numbers with the same exponents. Total mass = (5.97 × 10^2 × 10^22) + (7.35 × 10^22) = (597 + 7.35) × 10^22 = 604.35 × 10^22 kg.

To find the total mass of the Earth and Moon combined, we write both masses in scientific notation. The Earth’s mass is 5.97 × 10^24 kg and the Moon's is 7.35 × 10^22 kg. To add these, we first adjust the Earth's mass so that both terms have the same exponent (2 in this case). After reformatting, we easily add the two numbers before multiplying by the power of ten. This results in the total mass.

Think of this like adding money. If one person has $597 and another has$7.35, you can't easily just add them together unless you convert everything to cents (like 59700 cents and 735 cents) before adding. The same principle applies when dealing with numbers in scientific notation.

The distance between Sun and Earth is 1.496 × 10^11 m and distance between Earth and Moon is 3.84 × 10^8 m. Distance between Sun and Moon = 1.496 × 10^11 m - 3.84 × 10^8 m. = 1.496 × 10^11 - 3.84 × 10^8 = 1496 × 10^8 - 3.84 × 10^8 = (1496 - 3.84) × 10^8 m = 1492.16 × 10^8 m.

To determine the distance between the Sun and the Moon, we subtract the distance from the Earth to the Moon from the distance from the Earth to the Sun. Both distances are written in scientific notation. The first step is to rewrite both numbers so they share the same exponent, which allows for straightforward subtraction. This gives us the distance from the Sun to the Moon.

Consider this like calculating travel distances. If you know how far you are from your home to a friend's place (let's say 1.496 km) and from the friend's place to a restaurant (0.000384 km), you can find out how far the restaurant is from your home by subtracting. It’s much easier to visualize these numbers using scientific notation.

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